the mapping \(\Theta _{\omega }:U_{\omega }(x,r_0)\rightarrow C(B_{(\omega , x_{\omega })}^{u}(2\delta \ell _0^{-3}),E^{cs}(\omega ,x_{\omega }))\)  defined by \(\Theta _{\omega }(y)=g_{(\omega ,x_{\omega })}^{y}\)  is continuous with respect to the \(C^0\)  norm \(||\cdot ||\) .
With a minor modification of the proof of [1]}, we give the proof of this lemma in the following.
Let \((\omega ,x),(\omega ,y)\in \Gamma _{\ell _0}\) .
For ease notations, we denote by \(g_y:=g_{(\omega ,y)}\) , \(E^{\tau }_y:=E^{\tau }(\omega ,y)\) , \(\pi ^{\tau }_{y}:=\pi ^{\tau }_{(\omega ,y)}\)  and
\(W^{u}_{x,\delta ,y}:=\exp ^{-1}_x \exp _y \operatorname{graph}(g_y|_{\widetilde{B}^{u}_{(\omega ,y)}(\delta \ell ^{-2}_{0})})\)  for \(\tau =u,cs\)  and \(\delta >0\) .
We first show that, \(\pi ^{u}_{x}\)  is invertible on \(W^{u}_{x,4\delta ,y}\)  when \(|x-y|\)  and \(\delta \)  are small enough.
This is guaranteed if for all \(u_1, u_2\in W^{u}_{x,4\delta ,y}\) , one has \(|\pi ^{cs}_{x}(u_1-u_2)|<|u_1-u_2|\) .
By Theorem REF , for every \(u_1, u_2\in W^{u}_{x,4\delta ,y}\)  we have
\(|\pi ^{cs}_{x}(u_1-u_2)|\le (|\pi ^{cs}_x|_{E^{u}_y}|+|\pi ^{cs}_x|_{E^{cs}_y}|\cdot \operatorname{Lip}(g_y))\cdot |u_1-u_2|,\) 
where \(\operatorname{Lip}(g_y)\)  is the Lipschitz constant of \(g_y\)  with respect to the norm \(|\cdot |\) .
We show that \(\operatorname{Lip}(g_y)\ll 1\)  when \(\delta \)  small enough as follows: for every \(u_1,u_2\in W^{u}_{x,4\delta ,y}\) ,
\(|g_yu_1-g_yu_2|&\le 3 |g_yu_1-g_yu_2|^{\prime }_{(\omega ,y)}\\&\le 3{\operatorname{Lip}}^{\prime }_{y}(g_y) \cdot |u_1-u_2|^{\prime }_{(\omega ,y)} \\&\le 3\ell _{0} {\operatorname{Lip}}^{\prime }_{y}(g_y) \cdot |u_1-u_2|\\&\le 3\ell _{0} (\sup _{v\in \widetilde{B}^{u}_{(\omega ,y)}(4\delta \ell _0^{-2})} |Dg_y (v)|^{\prime }_{}) \cdot |u_1-u_2|\\&\le 12\delta \ell _{0}^{-1} {\operatorname{Lip}}^{\prime }_{y}(Dg_y) \cdot |u_1-u_2|,\) 

where “\({\operatorname{Lip}}^{\prime }_{y}\) " is the Lipschitz constant with respect to the norm \(|\cdot |^{\prime }_{(\omega ,y)}\) .
By Theorem REF , we have \({\operatorname{Lip}}^{\prime }_{y}(Dg_y)\le C_0\ell _0\) .
Therefore, we have \(\operatorname{Lip}(g_y)\le 12C_0 \delta \)  which we may assume \(\ll 1\) .
By Lemma REF , we have \(|\pi ^{cs}_x|_{E^{u}_y}|\rightarrow 0, |\pi ^{cs}_x|_{E^{cs}_y}| \rightarrow 1\)  whenever
\(d_H(E^u_y,E^u_x)\rightarrow 0\)  and \(d_H(E^{cs}_y,E^{cs}_x)\rightarrow 0\) .
Thus, with Proposition REF , take \({r}^{\prime }_1(\ell _0, \omega )\)  small enough such that for every \(x, y\in U_{\omega }(x_{\omega },{r}^{\prime }_1)\) , \(d_H(E^u_y,E^u_x)\)  and \(d_H(E^{cs}_y,E^{cs}_x)\)  are sufficiently small so that
\(|\pi ^{cs}_x|_{E^{u}_y}|+|\pi ^{cs}_x|_{E^{cs}_y}|\cdot \operatorname{Lip}(g_y)<1.\) 
We define the function \(g^{y}_{(\omega ,x)}\)  by \(g^{y}_{(\omega ,x)}(u):=\pi ^{cs}_x((\pi ^{u}_x)^{-1}(u))\)  for \(u\in \pi ^{u}_x(W^{u}_{x,4\delta ,y})\) , where \((\pi ^{u}_x)^{-1}(u)\)  is the unique point in \(W^{u}_{x,4\delta ,y}\)  such that \(\pi ^{u}_x((\pi ^{u}_x)^{-1}(u))=u\) .
Take \(r_1(\ell _0,\delta ,\omega )<{r}^{\prime }_1\)  small enough, so that for every \(x, y\in U_{\omega }(x_{\omega },r_1)\)  one has
\(\pi ^u_x (\exp ^{-1}_x\circ \exp _y\operatorname{graph}(g_y|_{B^{u}_{(\omega ,y)}(\delta \ell _{0}^{-3})}))&\subset B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})\\&\subset \pi ^u_x(\exp ^{-1}_x\circ \exp _y\operatorname{graph}(g_y|_{B^{u}_{(\omega ,y)}(4\delta \ell _{0}^{-3})})).\) 

Then, by definition \(g^{y}_{(\omega ,x)}\)  is well-defined and continuous on \(B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})\) .
In particularly, for each \(y\in U_{\omega }(x_{\omega },r_1)\)  we consider the function \(g^{y}_{(\omega ,x_{\omega })}\)  which defined on \(B^{u}_{(\omega ,x_{\omega })}(2\delta \ell _0^{-3})\) , then by definition one has \(g^{y}_{(\omega ,x_{\omega })}\)  satisfying item (1).
For item (2), for each \(x\in U_{\omega }(x_{\omega },r_1)\) , let \(y^n\in U_{\omega }(x_{\omega },r_1), n\in \mathbb {N}\)  be a sequence with \(y^n \rightarrow x\)  as \(n\rightarrow \infty \) .
Shirking \(r_1(\ell _0,\delta ,\omega )\)  if necessary, such that for every \(x, y\in U_{\omega }(x_{\omega },r_1)\)  one has
\(\exp _{x_{\omega }} \operatorname{graph}(g^{y}_{(\omega ,x_{\omega })}|_{B^{u}_{(\omega ,x_{\omega })}(2\delta \ell _0^{-3})})\subset \exp _{x} \operatorname{graph}( g^{y}_{(\omega ,x)}|_{B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})}).\) 
Therefore, to prove \(||\Theta _{\omega }(y^n)-\Theta _{\omega }(x)||\rightarrow 0\)  as \(n\rightarrow \infty \) .
It suffices to prove \(||g^{y^n}_{(\omega ,x)}-g_{(\omega ,x)}||\rightarrow 0 \)  on \(B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})\)  as \(n\rightarrow \infty \) .
Let \(\gamma >0\)  be fixed, we prove that \(||g^{y^n}_{(\omega ,x)}-g_{(\omega ,x)}||<\gamma \)  for \(n\)  large enough.
For \(k\in \mathbb {N}\) , denote by \(x_{-k}=f^{-k}_{\omega }x\)  and \(y^n_{-k}=f^{-k}_{\omega }y^n\) .
For each \(n,k\in \mathbb {N}\) ,
let
\(\textbf {0}_{y^n_{-k}}: \widetilde{B}^{u}_{\Phi ^{-k}(\omega ,y^n)}({\delta }^{\prime }_1 \ell (\Phi ^{-k}(\omega ,y^n))^{-1})\rightarrow E^{cs}(\Phi ^{-k}(\omega ,y^n))\) 
be the function that is identically equal to 0, and let
\(\phi _{y^n_{-k}}:=\mathcal {T}_{\Phi ^{-1}(\omega ,y^n)}\circ \cdots \circ \mathcal {T}_{\Phi ^{-k}(\omega ,y^n)} \textbf {0}_{y^n_{-k}},\) 
where \(\mathcal {T}\)  is the graph transformation as in Lemma REF .
Likewise, define \(\phi _{x_{-k}}\) .
Note that for fixed \(k\in \mathbb {N}\) , we have \(y_{-k}^n, x_{-k}\in (\Gamma _{\ell _0e^{k\varepsilon _1}})_{\theta ^{-k}\omega }\)  for every \(n\in \mathbb {N}\) .
By Proposition REF , for fixed \(k\)  one has \(E^u(\theta ^{-k}\omega ,y_{-k}^n)\rightarrow E^u(\theta ^{-k}\omega ,x_{-k})\)  as \(n\rightarrow \infty \) .
This implies that
\(\exp _{y^n_{-k}} B^{u}_{\Phi ^{-k}(\omega ,y^n)}(3\delta \ell _0^{-3}) \rightarrow \exp _{x_{-k}} B^{u}_{\Phi ^{-k}(\omega ,x)}(3\delta \ell _0^{-3}) \ \text{as} \ n\rightarrow \infty ,\) 
and so, their \(f_{\theta ^{-k} \omega }^{k}\) -images as well.
Then, one can define the map \(\phi ^{y^n_{-k}}_{(\omega ,x)}\)  on \(B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})\)  as in item (1) when \(n\)  large enough and
\(||\phi _{x_{-k}}-\phi ^{y^n_{-k}}_{(\omega ,x)}||\rightarrow 0\)  on \(B^{u}_{(\omega ,x)}(3\delta \ell _0^{-3})\)  as \(n\rightarrow \infty \) .
Recall the uniform equivalence of the \(|\cdot |\)  and \(|\cdot |^{\prime }_{(\omega ,y)}\)  on \(\Gamma _{\ell _0}\) .
By Lemma REF , there exists a \(k_0\)  large enough such that for all \(n\)  large enough we have
\(||\phi ^{y^n_{-k_0}}_{(\omega ,x)}-g_{(\omega ,x)}^{y^n}||<\gamma /3 \ \text{and}\ ||\phi _{x_{-k_0}}-g_{(\omega ,x)}^{x}||<\gamma /3.\) 
To finish, we estimate \(||g_{(\omega ,x)}^{y^n}-g_{(\omega ,x)}^{x}||\)  by
\(||g_{(\omega ,x)}^{y^n}-g_{(\omega ,x)}^{x}||\le ||g_{(\omega ,x)}^{y^n}-\phi ^{y^n_{-k_0}}_{(\omega ,x)}||+||\phi ^{y^n_{-k_0}}_{(\omega ,x)}-\phi _{x_{-k_0}}||+||g_{(\omega ,x)}^{x}-\phi _{x_{-k_0}}||.\) 

Since the first and third terms above are \(<\gamma /3\) , we choose \(n\)  large enough such that the middle term is \(< \gamma /3\) .
This completes the proof.
We referred to sets of the form
\(\mathcal {S}(\omega ,x_{\omega })=\bigcup _{y\in \bar{U}_{\omega }} \exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y)),\) 
where \(x_{\omega },U_{\omega }(x_{\omega },r_0)\)  and \(\Theta \)  as in above and \(\bar{U}_{\omega }\subset U_{\omega }(x_{\omega },r_0)\)  is a compact subset,
as a unstable stack of local unstable manifolds through the point \((\omega ,x_{\omega })\) .
By Lemma REF  (2), the set \(\mathcal {S}(\omega ,x_{\omega })\)  is a compact subset of \(\mathcal {B}\) .
Lemma 4.7 
For fixed \((\omega ,x_{\omega })\in \Gamma _{\ell _0}\) . Let
\(\Theta _{\omega }\)  be defined in Lemma REF .
For \(y_i \in U_{\omega }(x_{\omega },r_1), i=1,2\)  with
\(\exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_1))\ne \exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_2)).\) 
Then, we have
\(\exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_1)) \cap \exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_2))=\varnothing .\) 

Denote by \(G_1:=\exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_1))\)  and \(G_2:=\exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y_2))\) .
If there exists \(z \in G_1\cap G_2\) , then \(z\)  contained in some local unstable manifolds of \((\omega ,y_1)\)  and \((\omega ,y_2)\) .
For each \({z}^{\prime }\in G_1\) , by Lemma REF  one has
\(\limsup _{n\rightarrow \infty }\frac{1}{n}\log |f_{\omega }^{-n}({z}^{\prime })- f_{\omega }^{-n}(y_2)|&\le \limsup _{n\rightarrow \infty } \frac{1}{n}\log (|f_{\omega }^{-n}({z}^{\prime })- f_{\omega }^{-n}(y_1)|\\+&|f_{\omega }^{-n}(z)- f_{\omega }^{-n}(y_1)|+|f_{\omega }^{-n}(z)- f_{\omega }^{-n}(y_2)|) \le -\frac{\lambda }{2}.\) 

Therefore, also by Lemma REF  one can show that \({z}^{\prime }\)  is contained in some local unstable manifold of \((\omega ,y_2)\) .
This implies that \({z}^{\prime }\in G_2\)  and so, \(G_1 \subset G_2\) .
Symmetrically, we also have \(G_1 \supset G_2\) .
This completes the proof.
Lemma 4.8 
Let \(\mathcal {S}(\omega ,x_{\omega })\)  be defined above, and let \(\Sigma _{(\omega ,x_{\omega })}:=\exp _{x_{\omega }}^{-1}\mathcal {S}(\omega ,x_{\omega }) \cap E^{cs}(\omega ,x_{\omega })\) .
Then \(\mathcal {S}(\omega ,x_{\omega })\)  is homeomorphic to
\(\Sigma _{(\omega ,x_{\omega })}\times B^{u}_{(\omega ,x_{\omega })}(2\delta \ell ^{-3}_0)\) 
under the mapping \(\Psi (\sigma ,u):=\exp _{x_{\omega }}(u+g^{\omega }_{\sigma }(u))\) ,
where \(g^{\omega }_{\sigma }=\Theta _{\omega }(y)\)  is the unique leaf of \(\mathcal {S}_{\omega }\)  such that \(\Theta _{\omega }(y)(0)=\sigma \) .

By Lemma REF , the mapping \(g^{\omega }_{\sigma }\)  is well defined.
Since \(\Psi \)  is a bijection from two compact sets, we only need to show that \(\Psi \)  is continuous.
This is guaranteed by the fact that \(\sigma \mapsto g^{\omega }_{\sigma }\)  is continuous, which in turn is a direct consequence of Lemma REF .

Random unstable stacks
We now construct the random unstable stacks, which is a random compact subset of \(\Omega \times \mathcal {B}\)  with positive \(\mu \) -measure, and for each \(w\)  fixed, the \(\omega \) -section \(\mathcal {S}_{\omega }\)  has a measurable partition which is consisted of local unstable manifolds.
Choose \(\ell _0\)  large enough for which \(\mu (\Gamma _{\ell _0})>0\)  and fixed \(\delta >0\)  sufficiently small.
By Proposition \(\ref {Prop:Trandom compact}\) , for each \(\epsilon >0\)  small enough there exists a compact subset \(A_{\epsilon }\subset \mathcal {B}\)  such that
\(\mathbb {P}\lbrace \omega :A_{\omega }\subset A_{\epsilon }\rbrace >1-\epsilon \) .
This implies that \(\mu (\Omega \times A_{\epsilon })>1-\epsilon \)  and so one can assume that
\(\mu (\Gamma _{\ell _0}\cap (\Omega \times A_{\epsilon }))>0.\) 
Let \(r_2\ll \delta \) , by compactness one can choose a point \(x_{\ast }\)  as a reference point satisfies that
\(\mu (\Gamma _{\ell _0}\cap (\Omega \times (B(x_{\ast },r_2)\cap A_{\epsilon }))>0,\) 
where \(B(x_{\ast },r_2):=\lbrace y:|y-x_{\ast }|\le r_2\rbrace \) .
Denote by \(A_{\ast }:= B(x_{\ast },r_2)\cap A_{\epsilon }\)  which is compact, and denote by \(W^{u}_{\delta ,(\omega ,y)}:=\exp _{y}\operatorname{graph}(g_{(\omega ,y)}|_{B^{u}_{(\omega ,y)}(\delta \ell _{0}^{-3})})\) .
Let
\(S_{\omega }:=\bigcup \lbrace W^{u}_{\delta ,(\omega ,y)}\cap B(x_{\ast },2r_2): y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace ,\) 
By chosen \(r_2\)  small enough (depend only on \(\delta , \ell _{0}\) ), for each \(y_i \in A_{\ast }\)  with \((\omega ,y_i)\in \Gamma _{\ell _0}\) , \(i=1,2\) ,
we can assume that: \(W^{u}_{\delta ,(\omega ,y_i)}\cap B(x_{\ast },2r_2)\)  is connected, and either
\(W^{u}_{\delta ,(\omega ,y_1)}\cap B(x_{\ast },2r_2)=W^{u}_{\delta ,(\omega ,y_2)}\cap B(x_{\ast },2r_2),\) 
or the two terms in the above equation are disjoint.
By Lemma REF , for fixed \(\omega \) , the subset \(\mathcal {S}_{\omega }\)  is compact, and the mapping \(y \mapsto W^{u}_{\delta ,(\omega ,x_y)}\cap B(x_{\ast },2r_2)\)  is continuous on \(\mathcal {S}_{\omega }\)  with respect to the Hausdorff distance between compact subsets of \(\mathcal {B}\) , where \(x_y\in A_{\ast }\cap (\Gamma _{\ell _{0}})_{\omega }\)  is the unique point such that \(y\in W^{u}_{\delta ,(\omega ,x_y)}\cap B(x_{\ast },2r_2)\) .
Indeed, we can choose a small number \(r_{1}(\omega )\) , such that for each \(x\in (\Gamma _{\ell _0})_{\omega }\) , the unstable stacks
\(\mathcal {S}(\omega ,x):=\bigcup _{y\in \bar{U}_{\omega }} \exp _{x}(\operatorname{graph}\Theta _{\omega }(y))\) 
be defined, where \(\bar{U}_{\omega }=A_{\ast }\cap U_{\omega }(x,r_1)\)  is a compact subset.
By the choice of \(r_2\ll \delta \)  one can assume that
\(W^{u}_{\delta ,(\omega ,x)}\cap B(x_{\ast },2r_2)=\exp _{x}(\operatorname{graph}\Theta _{\omega }(y))\cap B(x_{\ast },2r_2).\) 
Then, by compactness, there is a finitely subset \(\lbrace x_{\omega ,i}\rbrace ^{k(\omega )}_{i=1}\subset (\Gamma _{\ell _0})_{\omega } \cap A_{\ast }\) 
such that
\(\mathcal {S}_{\omega }= \bigcup ^{k(\omega )}_{i=1} \mathcal {S}(\omega , x_{\omega ,i})\cap B(x_{\ast },2r_2).\) 
Since for each \(i=1,\cdots ,k(\omega )\) , one has \(\mathcal {S}(\omega , x_{\omega ,i})\)  is compact, and so \(\mathcal {S}_{\omega }\)  is also compact.
By Lemma REF , it is easy to deduce that
\(y \mapsto W^{u}_{\delta ,(\omega ,x_y)}\cap B(x_{\ast },2r_2)\)  is continuous on \(\mathcal {S}(\omega , x_{\omega ,i})\)  for each \(i=1,\cdots ,k(\omega )\) , where \(x_y\in \bar{U}_{\omega }(x_{\omega ,i},r_1)\)  is the unique point such that \(y\in \exp _{x_{\omega ,i}}\operatorname{graph}(\Theta _{\omega }(x_y))\) .
So, the mapping is also continuous on \(\mathcal {S}_{\omega }\) .
We define the subset \(\mathcal {S}\)  of \(\Omega \times \mathcal {B}\)  by
\(\mathcal {S}:=\bigcup \lbrace \lbrace \omega \rbrace \times S_{\omega }: \exists y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace .\) 
Lemma 4.9 
The set \(\mathcal {S}\)  is random compact, and so is measurable with \(\mu (\mathcal {S})>0\) .

By Proposition REF , it suffices to show that, for every open set \(U\)  of \(\mathcal {B}\)  the set \(\lbrace \omega :S_{\omega }\cap U\ne \varnothing \rbrace \)  is measurable.
By the construction of \(\mathcal {S}\) , we have
\(\lbrace \omega :S_{\omega }\cap U\ne \varnothing \rbrace =\pi _{\Omega }\lbrace (\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast }):W^{u}_{\delta ,(\omega ,x)}\cap B(x_{\ast },2r_2)\cap U\ne \varnothing \rbrace ,\) 
where \(\pi _{\Omega }\)  be the canonical projection from \(\Omega \times \mathcal {B}\)  onto \(\Omega \) .
So, by Proposition REF  we only need to show that
\(\lbrace (\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast }): W^{u}_{\delta ,(\omega ,x)}\cap B(x_{\ast },2r_2)\cap U\ne \varnothing \rbrace \) 
is a measurable subset of \(\Omega \times \mathcal {B}\) .
Let \(m:= \dim E^{u}\) , and let \(\lbrace v_k\rbrace _{k>0}\)  be a countable dense subset of \(\mathbb {R}^{m}\) .
Choose a measurable basis \(\lbrace e_i{(\omega ,x)}\rbrace _{i=1}^{m}\)  spanning \(E^{u}(\omega ,x)\) , for each \(v_k=(v_k^i)\in \mathbb {R}^{m}\) , let \(v_k(\omega ,x):= \sum _{i}v_k^i e_i{(\omega ,x)}\) .
Then, the mapping \((\omega ,x)\mapsto v_k(\omega ,x)\)  is measurable for every \(k\) .
By density, restricted to the subset \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\)  we have
\(\lbrace (\omega ,x): W^{u}_{\delta ,(\omega ,x)}\cap B(x_{\ast },2r_2)\cap U\ne \varnothing \rbrace =\bigcup _{k>0}(\lbrace (\omega ,x): |v_k(\omega ,x)|\le \delta \ell ^{-3}_0\rbrace \bigcap &\\\lbrace (\omega ,x): v_k(\omega ,x)+g_{(\omega ,x)}v_k(\omega ,x)+x\in B(x_{\ast },2r_2)\cap U\rbrace &).\) 

By Remark REF , the map \((\omega ,x) \mapsto g_{(\omega ,x)}v_k(\omega ,x)\)  is measurable on the measurable subset
\(\lbrace (\omega ,x): |v_k(\omega ,x)|\le \delta \ell ^{-3}_0\rbrace \)  for every \(k\) .
Therefore, one has
\(\lbrace (\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast }): W^{u}_{\delta ,(\omega ,x)}\cap B(x_{\ast },2r_2)\cap U\ne \varnothing \rbrace \) 
is measurable.
The construction of \(\mathcal {S}\)  implies that \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\)  is a subset of \(\mathcal {S}\) , so one has \(\mu (\mathcal {S})>0\) .
For each \(\omega \) , let \(\xi _{\omega }\)  be a partition of \(\mathcal {S}_{\omega }\)  set by
\(\xi _{\omega }:=\lbrace W^{u}_{\delta ,(\omega ,y)}\cap B(x_{\ast },2r_2): y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace .\) 
For each \(W \in \xi _{\omega }\) , we call \(W\)  is a unstable leave of \(\mathcal {S}_{\omega }\) .
Since \(S_{\omega }\)  is contained in the finite union of unstable stacks of local unstable manifolds.
By Lemma REF , one can easily deduced that \(\eta _{\omega }\)  is a measurable partition of \(\mathcal {S}_{\omega }\) .
We call the set \(\mathcal {S}\)  as a random unstable stack with the center \(x_{\ast }\) .

Properties of maps on unstable stacks
Let the random unstable stack \(\mathcal {S}\)  be defined above.
Lemma 4.10 
For all \(n\in \mathbb {N}\) , the map \(f^{-n}_{\omega }\)  is defined and continuous on \(S_{\omega }\) .

For each \(n\in \mathbb {N}\) , the map \(f^{-n}_{\omega }\)  is defined on \(S_{\omega }\)  follows from Theorem REF .
We now prove it is continuous.
Let \(r_1(\omega )\)  be a small number satisfying Lemma REF , so that for each \(x\in (\Gamma _{\ell _0})_{\omega }\)  the unstable stacks
\(\mathcal {S}(\omega ,x):=\bigcup _{y\in \bar{U}} \exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y))\) 
is defined, where \(\bar{U}:=A_{\ast }\cap U_{\omega }(x,r_1)\) .
Since there is a finitely subset \(\lbrace x_{\omega ,i}\rbrace ^{k(\omega )}_{i=1}\subset (\Gamma _{\ell _0})_{\omega } \cap A_{\ast }\)  such that
\(\mathcal {S}_{\omega }\subset \bigcup ^{k(\omega )}_{i=1} (\mathcal {S}(\omega , x_{\omega ,i})\cap B(x_{\ast },2r_2)).\) 
It suffices to show \(f^{-n}_{\omega }\)  is continuous on \(\mathcal {S}(\omega ,x)\)  for every \(x\in (\Gamma _{\ell _0})_{\omega } \cap A_{\ast }\) .
We claim that, by shrinking \(r_1>0\) , there exists \(n_0\in \mathbb {N}\) , for each \(n\ge n_0\)  there exists a compact subset \(V_{n,x}\subset (\Gamma _{\ell _0e^{n\varepsilon _1}})_{\theta ^{-n}\omega } \)  such that one can define a compact unstable stack \(\mathcal {S}_{n,x}\)  of local unstable manifolds with \(f^{n}_{\theta ^{-n}\omega }(\mathcal {S}_{n,x})\supset \mathcal {S}(\omega ,x)\) .
Assuming the claim holds. For \(n\ge n_0\) , by \(f^{n}_{\theta ^{-n}\omega }\)  is injective, one has \(f^{-n}_{\omega }\)  is continuous on \(\mathcal {S}(\omega ,x)\) . For \(n<n_0\) , only need to write \(f^{n}_{\omega }=f^{n_0-n}_{\theta ^{-{n_0}}\omega } f^{-n_0}_{\omega }\) .
It remains to show that the claim holds.
Note that \(f^{-n}_{\omega } \bar{U}\subset (\Gamma _{\ell _0e^{n\varepsilon _1}})_{\theta ^{-n}\omega }\) ,
so \(z\mapsto E^{u}(\theta ^{-n}\omega ,z)\)  and \(z\mapsto E^{cs}(\theta ^{-n}\omega ,z)\)  are continuous on \(f^{-n}_{\omega } \bar{U}\) .
Shrinking \(r_1>0\) , such that for every \(z\in f^{-n}_{\omega } \bar{U}\) , one has \(d_H(E^{cs}(\theta ^{-n}\omega ,z), E^{cs}(\theta ^{-n}\omega ,f_{\omega }^{-n}x))\)  and
\(d_H(E^{u}(\theta ^{-n}\omega ,z), E^{u}(\theta ^{-n}\omega ,f_{\omega }^{-n}x))\)  small enough,
so that the map \(g^{f_{\omega }^{-n}z}_{\Phi ^{-n}(\omega ,x)}\)  as in Lemma REF  is defined on \(D_{n,x}\) ,
where \(D_{n,x}:=B^{u}_{\Phi ^{-n}(\omega ,x)}(2\delta (e^{n\varepsilon _1}\ell _0)^{-3})\) .
Let \(V_{n,x}:=f^{-n}_{\omega } \bar{U}\) , and let
\(\mathcal {S}_{n,x}=\bigcup _{z\in V_{n,x}} \exp _{f^{-n}_{\omega }x} \operatorname{graph}(g^{f_{\omega }^{-n}z}_{\Phi ^{-n}(\omega ,x)}|_{D_{n,x}}).\) 
Shrinking \(r_1>0\)  if necessary, one may assume that for every \(z\in V_{n,x}\)
\(\exp _{z} \operatorname{graph}(g_{(\theta ^{-n}\omega ,z)}|_{\widetilde{B}^{u}_{(\theta ^{-n}\omega ,z)}(\frac{1}{6}\delta (e^{n\varepsilon _1}\ell _0)^{-3})})\subset \exp _{f^{-n}_{\omega }x} \operatorname{graph}(g^{f_{\omega }^{-n}z}_{\Phi ^{-n}(\omega ,x)}|_{D_{n,x}}).\) 
By Theorem REF  (iii) and Lemma REF  (1), we can choose \(n_0\)  large enough (depends only on \(\ell _0\) ), such that for every \(n\ge n_0\)
\(& \widetilde{f}^{n}_{(\theta ^{-n}\omega ,z)}\operatorname{graph}(g_{(\theta ^{-n}\omega ,z)}|_{\widetilde{B}^{u}_{(\theta ^{-n}\omega ,z)}(\frac{1}{6}\delta (e^{n\varepsilon _1}\ell _0)^{-3})})\\\supset & \operatorname{graph}(g_{(\omega ,f^{n}_{\theta ^{-n}\omega }(z))}|_{\widetilde{B}^{u}_{(\omega ,f^{n}_{\theta ^{-n}\omega }(z))}(4\delta \ell _{0}^{-2})})\\\supset & \operatorname{graph}\Theta _{\omega }(f^{n}_{\theta ^{-n}\omega }(z)).\) 

Therefore, we have \(f^{n}_{\theta ^{-n}\omega }\mathcal {S}_{n,x}\subset \mathcal {S}(\omega ,x)\)  for each \(n\ge n_0\) .
Let \(\xi _{\omega }\)  be the measurable partition of \(\mathcal {S}_{\omega }\)  into unstable leaves.
Let \(\xi :=\lbrace \lbrace \omega \rbrace \times W:W\in \xi _{\omega }\rbrace \)  be a partition of \(\mathcal {S}\) , since \(\Omega \)  is Polish space one has \(\xi \)  is also measurable.
For each \((\omega ,x)\in \mathcal {S}\) , denote by \(\xi (\omega ,x)\)  the element of \(\xi \)  which contained \((\omega ,x)\)  and denote by \(\xi (\omega ,x)_{\omega }\)  the \(\omega \) -section of the subset \(\xi (\omega ,x)\)  of \(\Omega \times \mathcal {B}\) .
Then by construction one has \(\xi _{\omega }(x)=\xi (\omega ,x)_{\omega }:=\lbrace y:(\omega ,y)\in \xi (\omega ,x)\rbrace \) .
Recall the induced volume \(\nu _{(\omega ,x)}\)  which we defined in Section REF .
Lemma 4.11 
For any measurable set \(E \subset \Omega \times \mathcal {B}\) , the map \( (\omega ,x) \mapsto \nu _{(\omega ,x)}((E \cap \xi (\omega ,x))_{\omega })\)  is measurable on \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) .

Let \(m:=\dim E^{u}\) , choose a measurable basis \(\lbrace e_i{(\omega ,x)}\rbrace _{i=1}^{m}\)  spanning \(E^{u}(\omega ,x)\) .
For each \(t=(t_1,\cdots ,t_m)\in \mathbb {R}^{m}\) , let \(I_{(\omega ,x)}(t)=\sum _{i=1}^{m}t_ie_i{(\omega ,x)}\) .
Then, for
\((\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) 
we have
\(\nu _{(\omega ,x)}((E \cap \xi (\omega ,x))_{\omega }):=\int _{E_{(\omega ,x)}}\det (Id+Dg_{(\omega ,x)}(I_{(\omega ,x)}(t)))dt.\) 
Where \(E_{(\omega ,x)}:=\lbrace t\in \mathbb {R}^{m}: I_{(\omega ,x)}(t)+g_{(\omega ,x)}(I_{(\omega ,x)}(t))+x\in E_{\omega }\cap B(x_{\ast },2r_2)\rbrace \cap \lbrace t\in \mathbb {R}^{m}:|I_{(\omega ,x)}(t)|\le \delta \ell ^{-3}_0\rbrace \) 
and the “\(\det \) " with respect to the Euclidean volume on \(\mathbb {R}^{m}\)  and the induced volume \(m_{T_{I_{(\omega ,x)}(t)} \widetilde{W}^{u}_{\delta }(\omega ,x)}\) .
Since \(\Omega \times \mathcal {B}\)  is a Polish space, without loss of generality, we can assume \(E\)  is a compact set.
For fixed \(t\in \mathbb {R}^{m}\)  with \(|I_{(\omega ,x)}(t)|\le \delta \ell ^{-3}_0\) .
By Remark REF , we have \((\omega ,x)\mapsto \chi _{E(\omega ,x)}(t)\)  is a measurable function on \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) ,
and the similar discussion in Lemma REF  shows that \((\omega ,x)\mapsto \det (Id+Dg_{(\omega ,x)}(I_{(\omega ,x)}(t)))\) 
is also measurable on \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) .
Therefore, for every \(M\ge 0\) , the set
\(E_M:=\lbrace (\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast }):\nu _{(\omega ,x)}((E \cap \xi (\omega ,x))_{\omega })\le M\rbrace \) 
is a measurable subset of \(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) .
In the remainder of this section, we assume that \(E^{c}(\omega ,x)=0\)  for every \((\omega ,x)\in \Gamma \) .
Lemma 4.12 
Assume \(E^{c}=\lbrace 0\rbrace \) . For each \(q\in \mathbb {N}\) , the map \(x\mapsto E^{u}(\omega ,x)\)  is continuous on \(f^{-q}_{\omega } S_{\omega }\)

We give the proof for \(q=0\) , and the case for \(q\ge 1\)  can be proven similarly.
Let \((\omega ,x)\in \mathcal {S}\)  be fixed, the main difference with the argument in Proposition REF  is that no intrinsically defined the stable subspace \(E^{cs}(\omega ,x)\) .
As in [1]}, we use the backward graph transform to overcome it.
For \((\omega ,x)\in \mathcal {S}\) , there exists a point \((\omega ,x_0)\in \Gamma _{\ell _0}\)  with the property that \(x\in W^{u}_{\delta ,(\omega ,x_0)}\) .
We write \(\widetilde{f}_{-n}:=\widetilde{f}_{\Phi ^{-n}(\omega ,x_0)}\) , \(|\cdot |^{\prime }_{-n}:=|\cdot |^{\prime }_{\Phi ^{-n}(\omega ,x_0)}\)  and
\({x}^{\prime }_{-n}:=f_{\omega }^{-n}x-f_{\omega }^{-n}x_{0}\)  for each \(n\ge 0\) .
Let \(F_0=E^{s}(\omega ,x_0)\) , then \(\mathcal {B}=E^{u}(\omega ,x)\oplus F_0\) .
Identifying \(\mathcal {B}_x\)  with \(\mathcal {B}_{x_0}\) . For \(\delta \)  small enough, by Lemma REF , we may assume that \(D\widetilde{f}_{-n}({x}^{\prime }_{-n})\)  is sufficiently close to \(D\widetilde{f}_{-n}(0)\)  so that the backward graph transform (Lemma REF ) can be applied to the linear map \(D\widetilde{f}_{-n}({x}^{\prime }_{-n})\)  for every \(n\ge 0\) .
Then, by Lemma REF  we get a sequence of subspace \(F_n\subset \mathcal {B},n\in \mathbb {N}\) , such that \(D\widetilde{f}_{-n}({x}^{\prime }_{-n})(F_n)\subset F_{n-1}\)  and for each \(v_{-n}\in F_n\) , one has
\(|D\widetilde{f}_{-n}({x}^{\prime }_{-n})v_{-n}|^{\prime }_{-n+1}\le (e^{-\lambda }+\delta ) |v_{-n}|^{\prime }_{-n}.\) 
Since \(F_n\)  is a graph of some linear map from \(E^{s}(\omega ,x_0)\)  to \(E^{u}(\omega ,x_0)\) , one has \(\mathcal {B}= E^{u}(\Phi ^{-n}(\omega ,x))\oplus F_{n}\) .
For every \(y_k\in \mathcal {S}_{\omega }\)  with \(y_k\rightarrow x\)  as \(k\rightarrow \infty \) . Let \(v^k\in E^{u}(\omega ,y_k)\)  and decompose \(v^k=v^{k,u}+v^{k,s}\)  with respect to the splitting \(\mathcal {B}=E^{u}(\omega ,x)\oplus F_0\) .
Using the same estimation in Proposition REF  one has \(v^{k,s}\rightarrow 0\)  as \(k\rightarrow \infty \) .
Hence, we have \(G(E^{u}(\omega ,y_k),E^{u}(\omega ,x))\rightarrow 0\)  as \(k\rightarrow \infty \) .
By Lemma REF , this is enough to show that \(x\mapsto E^{u}(\omega ,x)\)  is continuous on \(\mathcal {S}_{\omega }\) .
Let \(J^{u}(\omega ,x)=\det (Df_{\omega }(x)|E^u(\omega ,x))\)  for \((\omega ,x)\in \mathcal {S}\) .
By Proposition REF  (b) the following function is well defined on \(\xi _{\omega }(x)\)  and Lipschitz-continuous:
\(z\mapsto \Delta ^{\omega }(x,z):= \prod _{n=1}^{\infty } \dfrac{J^{u}(\Phi ^{-n}(\omega ,x))}{J^{u}(\Phi ^{-n}(\omega ,z))}, \quad z\in \xi _{\omega }(x).\) 
Define the function \(q(\omega ,z):\mathcal {S}\rightarrow \mathbb {R}\)  by
\(q(\omega ,z):=\dfrac{\Delta ^{\omega }(x,z)}{\int _{\xi _{\omega }(x)} \Delta ^{\omega }(x,y) d\nu _{(\omega ,x)}(y)}, \quad z\in \xi _{\omega }(x).\) 
Since for every \({x}^{\prime }\in \mathcal {B}\)  satisfied that \(\xi _{\omega }(x)=\xi _{\omega }({x}^{\prime })\) , one has
\(\Delta ^{\omega }({x}^{\prime },z)=\Delta ^{\omega }(x,z) \cdot \Delta ^{\omega }({x}^{\prime },x),\) 
so the function \(q\)  is not dependent on the choice of \(x\in \xi _{\omega }(z)\)  and is well defined as a function of \((\omega ,z)\) .
Moreover, Lemma REF  and Lemma REF  imply that for each fixed \(z\) , the map \(\omega \mapsto q(\omega ,z)\)  is measurable on the measurable set
\(\lbrace \omega :(\omega ,z)\in \mathcal {S}\rbrace \) .
Lemma 4.13 
Assume \(E^{c}=\lbrace 0\rbrace \) . For fixed \(\omega \) , the function \(q(\omega ,z):\mathcal {S}_{\omega }\rightarrow \mathbb {R}\)  is continuous.

We follow the proof of Lemma 7.9 in [1]}.
Let \(\omega \)  be fixed.
For each \(n\) , by Lemma REF  we have \(f_{\omega }^{-n}|_{\mathcal {S}_{\omega }}\)  is continuous,
by Lemma REF  we have
\(x\mapsto E^{u}(\omega ,x)\)  is continuous on \(f_{\omega }^{-n}{\mathcal {S}_{\omega }}\) .
Then, one has \(x\mapsto \det (Df_{\theta ^{-n}\omega }(f_{\omega }^{-n}x)|E^u(\Phi ^{-n}\omega ,x))\)  is continuous on \(\mathcal {S}_{\omega }\)  by Proposition REF .
Let \(r_1(\omega )\)  be a small number satisfying Lemma REF .
For each \((\omega ,x)\in \Gamma _{\ell _0}\cap (\Omega \times A_{\ast })\) .
Define the unstable stack \(\mathcal {S}(\omega ,x)\)  by
\(\mathcal {S}(\omega ,x):=\bigcup _{y\in \bar{U}} \exp _{x_{\omega }}(\operatorname{graph}\Theta _{\omega }(y))\) 
where \(\bar{U}=A_{\ast }\cap U_{\omega }(x,r_1)\) .
Since \(\mathcal {S}_{\omega }\)  is contained in the finite union of unstable stacks.
It suffices to show that \(q(\omega ,\cdot )\)  is continuous on \(\mathcal {S}(\omega ,x)\cap B(x_{\ast },2r_2)\) .
By Lemma REF , there exists a homeomorphism
\(\Psi _{\omega ,x}: \Sigma _{(\omega ,x)}\times B^{u}_{(\omega ,x)}(2\delta \ell ^{-3}_0)\rightarrow \mathcal {S}(\omega ,x)\) ,
where \(\Sigma _{(\omega ,x)}\)  as in Lemma REF .
For every \(z\in \mathcal {S}(\omega ,x)\cap B(x_{\ast },2r_2)\) , denote by \((\sigma _z,u_z):=\Psi ^{-1}_{\omega ,x}(z)\)  and let \(\sigma (z):=\Psi _{\omega ,x}(\sigma _z,0)\) .
Define
\(z\mapsto \bar{\Delta }_n(z):=\Delta ^{\omega }_{n}(\sigma (z),z),\) 
where \(\Delta ^{\omega }_{n}(\sigma (z),z)\)  as in \((\ref {eq:Delta n})\) .
Since \(\sigma (z)\)  depend continuously on \(z\) , one has \(z\mapsto \bar{\Delta }_n(z)\)  is continuous.
By Proposition REF ,
the sequence \(\bar{\Delta }_n\)  converges uniformly to \(\bar{\Delta }\)  which is continuous on \(\mathcal {S}(\omega ,x)\) .
It remains to show that \(z\mapsto \int \bar{\Delta } dv_{(\omega ,z)}\)  is continuous on \(\mathcal {S}(\omega ,x)\cap B(x_{\ast },2r_2)\) .
Denote by \(E_z:=\pi ^{u}_{(\omega ,x)}(\exp ^{-1}_{x}\xi _{\omega }(z))\) , since \(z\mapsto \xi _{\omega }(z)\)  is continuous,
we have \(z\mapsto E_z\)  is also continuous with respect to the Hausdorff topology.
By Lemma REF  and the change of variables formula we have
\(\int _{\xi _{\omega }(\sigma (z))} \bar{\Delta }(z) dv_{(\omega ,z)}=\int _{B^{u}_{(\omega ,x)}(2\delta \ell ^{-3}_0)}\bar{\Delta }(z) \det (Id+Dg_{\sigma _z}(u))\chi _{E_z}(u)dm(u)\) 
where \(m\)  is the induced volume on \(E^{x}\)  and the “\(\det \) " is with respect to \(m\)  and \(m_{E^u(\omega ,z)}\) .
By Lemma REF  we have that
\((\sigma ,u) \mapsto E^u(\omega ,\Psi _{\omega ,x}(\sigma ,u))\) 
is continuous on
\(\Sigma _{\omega ,x} \times B^{u}_{\omega ,x}(2\delta \ell ^{-3}_0)\) .
So, by Lemma REF  the map \((\sigma ,u) \mapsto Id+Dg_{\sigma }(u)\)  is continuous.
From this and from properties of the “det"-function (see Proposition REF ),
we deduce that the mapping \(z \mapsto \det (Id+g_{\sigma _z}(u))\)  is continuous for each \(u\) .
Therefore, one has \(z\mapsto \int \bar{\Delta } dv_{(\omega ,z)}\)  is continuous. This completes the proof.

Proof of main Theorem
In this section, we use the same method in [4]} (also in [1]} and [6]}) to give a detailed proof of Theorem REF  and Theorem REF  when \(\mu \)  is ergodic.
We recall some notations about partitions at the beginning.
Let \(T: X\rightarrow X\)  be a measure preserving transformation of a probability space \((X,\mu )\) .
For a measurable partition \(\eta \)  on \(X\) , we let \(\eta (x)\)  denote the atom of \(\eta \)  containing \(x\)  for each \(x\in X\) .
For two measurable partitions \(\eta _1\)  and \(\eta _2\)  , we say \(\eta _2\)  is a refinement of \(\eta _1\)  if \(\eta _1(x)\supset \eta _2(x)\)  for \(\mu \) -almost every \(x\) , and written by \(\eta _1\le \eta _2\) .
Let \( \eta _1 \vee \eta _2=\lbrace A\cap B:A\in \eta _1, B\in \eta _2\rbrace \)  and let \(T^{-1} \eta _1=\lbrace T^{-1}(A):A\in \eta _1\rbrace \) . We say \(\eta \)  is \(T\) -decreasing if \(\eta \le T^{-1} \eta \) .
We fixed \(\ell _0\)  large enough such that \(\mu (\Gamma _{\ell _0})>0\) , and let the random stack \(\mathcal {S}\)  as defined in Section REF .
Partitions subordinate to the unstable foliation
Let \(\xi _{\omega }\)  be the measurable partition of \(\mathcal {S}_{\omega }\)  into unstable leaves, and let \(\xi :=\lbrace \lbrace \omega \rbrace \times W_{\omega }:W_{\omega }\in \xi _{\omega }\rbrace \)  be the measurable partition of \(\mathcal {S}\) .
By definition, for each \((\omega ,x)\in \mathcal {S}\) , one has \(\xi (\omega ,x)\subset \lbrace \omega \rbrace \times \mathcal {B}\) .
We denote by \(\xi (\omega ,x)_{\omega }\)  the \(\omega \) -section of the subset \(\xi (\omega ,x)\)  of \(\Omega \times \mathcal {B}\) .
Denote by \(\lbrace \mu _{\xi (\omega ,x)}\rbrace _{(\omega ,x)\in \mathcal {S}}\)  the canonical disintegration of \(\mu |_{\mathcal {S}}\) ,
which satisfied that,
\(\mu |_{\mathcal {S}}=\int _{\mathcal {S}} \mu _{\xi (\omega ,x)} d\mu (\omega ,x).\) 
The family of measures \(\lbrace \mu _{\xi (\omega ,x)}\rbrace \)  on \(\mathcal {B}\)  (one can identify \(\lbrace \omega \rbrace \times \mathcal {B}\)  with \(\mathcal {B}\) ) exists by Rokhlin's disintegration theorem, and any two families like that are equal on a full \(\mu |_{\mathcal {S}}\) -measure subset of \(\mathcal {S}\) .
We give a formal definition of SRB measures for random dynamical systems on Banach spaces as follows:
Definition 5.1 
We say \(\mu \)  is a random SRB measure if (i) it has a positive exponent for \(\mu \) -almost every \((\omega ,x)\) , and (ii) for every random stack \(\mathcal {S}\)  with \(\mu (\mathcal {S})>0\) , \(\mu _{\xi (\omega ,x)}\)  is absolutely continuous with respect to \(\nu _{(\omega ,x)}\)  for \(\mu \) -almost every \((\omega ,x)\)  in \(\mathcal {S}\) .

Definition 5.2 
We say that a measurable partition \(\eta \)  is subordinate to the unstable foliation \(W^{u}\)  if for \(\mu \) -almost every \((\omega ,x)\)  we have

\(\eta (\omega ,x)\subset \lbrace \omega \rbrace \times W^{u}(\omega ,x)\) ,

\(\eta (\omega ,x)_{\omega }\)  contains a neighborhood of \(x\)  in \(W^{u}(\omega ,x)\) , and

\(\eta (\omega ,x)_{\omega } \subset f^{N}_{\theta ^{-n}\omega } W^{u}_{\operatorname{loc}}(\Phi ^{-n}(\omega ,x))\)  for some \(N\in \mathbb {N}\)  depending on \((\omega ,x)\) .


Lemma 5.1 The following are equivalent:

\(\mu \)  is a random SRB measure;

there exists a partition \(\eta \)  subordinate to \( W^{u}(\omega ,x) \)  with \(\mu _{\eta (\omega ,x)}\ll \nu _{(\omega ,x)}\)  for \(\mu \) -almost every \((\omega ,x)\) ;

for every partition \(\eta \)  subordinate to \( W^{u}(\omega ,x) \) , one has \(\mu _{\eta (\omega ,x)}\ll \nu _{(\omega ,x)}\)  for \(\mu \) -almost every \((\omega ,x)\) .


We first prove (2) implies (3): Let \(\eta _1\)  be the partition satisfies (2), for each \(\eta _2\)  subordinate to
\(W^{u}(\omega ,x)\) , we denote by \(\eta =\eta _1 \vee \eta _2\)  and let \(\lbrace \mu _{W}\rbrace _{W\in \eta }\)  be the canonical disintegration of \(\mu \)  with respect to \(\eta \) . Note that \(\mu _{\eta _1(\omega ,x)}\ll \nu _{(\omega ,x)}\)  implies that \(\mu _{\eta (\omega ,y)} \ll \nu _{(\omega ,x)}\)  for every \(\eta (\omega ,y) \subset \eta _1(\omega ,x)\) . Therefore, by the uniqueness of the canonical disintegration, one has \(\mu _{\eta _2(\omega ,x)}\ll \nu _{(\omega ,x)}\)  for \(\mu \) -almost every \((\omega ,x)\) .
The statement (1) implies (2) and, (3) implies (1) follow immediately from the proposition below.
Proposition 5.1 
Assuming that \((\Phi ,\mu )\)  has a positive Lyapunov exponent, then there is a random stack \(\mathcal {S}\)  with positive \(\mu \) -measure, and a measurable partition \( \eta \)  on \(\widetilde{\mathcal {S}}:=\cup _{n\ge 0}\Phi ^n \mathcal {S}\)  such that

\(\eta \)  subordinate to \( W^{u}(\omega ,x) \) ,

\(\eta \)  is \(\Phi \) -decreasing,

for any measurable set \(E \subset \Omega \times \mathcal {B}\) , the function \((\omega ,x) \mapsto \nu _{(\omega ,x)}(E_{\omega } \cap \eta (\omega ,x)_{\omega })\)  is finite-valued and measurable on a full \(\mu \) -measure set, and

\(\vee _{n\ge 0}\Phi ^{-n}\eta \)  is partition of \(\widetilde{\mathcal {S}}\)  into points modulo \(\mu \) -measure 0.


Using the notations in Section REF , we first construct a random stack.
Let \(r\in (1/2,1)\)  and let
\(\mathcal {S}^{r}_{\omega }:=\bigcup \lbrace W^{u}_{\delta ,(\omega ,y)}\cap \bar{B}(x_{\ast },2rr_2): y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace ,\) 
and define \(\mathcal {S}^{r}:=\bigcup \lbrace \lbrace \omega \rbrace \times S^{r}_{\omega }: \exists y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace \) .
Let \(\widetilde{\mathcal {S}}:=\widetilde{\mathcal {S}}^r=\cup _{n\ge 0}\Phi ^n \mathcal {S}^r\) , since \(\mu \)  is \(\Phi \) -ergodic one has \(\mu (\widetilde{\mathcal {S}})=1\) .
Let \(\xi :=\lbrace \lbrace \omega \rbrace \times W_{\omega }:W_{\omega }\in \xi _{\omega }\rbrace \)  be a measurable partition of \(\mathcal {S}^r\) , where \(\xi _{\omega }\)  is the measurable partition of \(\mathcal {S}^r_{\omega }\)  into unstable leaves.
For each \(n\ge 0\) , let \(\xi ^n=\lbrace \Phi ^n(W):W\in \xi \rbrace  \cup \lbrace \widetilde{\mathcal {S}}-\Phi ^n(\mathcal {S}^r)\rbrace \)  be a measurable partition of \(\widetilde{\mathcal {S}}\) , and let \(\eta =\vee _{n=0}^{\infty }\xi ^n\) .
Note that \(\eta \)  is a refinement of the partition \(\eta _0:=\lbrace \lbrace \omega \rbrace \times \mathcal {B}\rbrace _{\omega \in \Omega }\) .
For every \((\omega ,x)\in \widetilde{\mathcal {S}}\) , a point \((\omega ,y)\in \eta (\omega ,x)\)  if and only if
\(\left\lbrace \begin{alignedat}{2}&f^{-n}_{\omega } y \in \xi _{\theta ^{-n}\omega }(f^{-n}_{\omega } x)\quad &\text{if}& \ \Phi ^{-n}(\omega ,x) \in \mathcal {S}^r; \\&\Phi ^{-n}(\omega ,y) \notin \mathcal {S}^r &\text{if}& \ \Phi ^{-n}(\omega ,x) \notin \mathcal {S}^r.\end{alignedat}\right. \) 
Therefore, one has that \(\eta \)  is \(\Phi \) -decreasing. This proves item (b).
For each \((\omega ,x)\in \widetilde{\mathcal {S}}\) , there exists \(N(\omega ,x)\in \mathbb {N}\)  such that \(\Phi ^{-N}(\omega ,x) \in \mathcal {S}^r\) .
Then, by (REF ) we have \(\eta (\omega ,x)_{\omega }\in f^{N}_{\theta ^{-n}(\omega )} W^{u}_{\operatorname{loc}}(\Phi ^{-N}(\omega ,x))\) .
To show \(\eta \)  subordinate to \(W^{u}\) , we only requirement that \(\eta (\omega ,x)_{\omega }\)  contains a neighborhood of \(x\)  in \(W^{u}_{\operatorname{loc}}(\omega ,x)\)  for \(\mu \) -almost every \((\omega ,x)\)  left.
We now choose \(r\)  carefully to guarantee that.
Let
\( \partial \mathcal {S}^{r}_{\omega }:=\bigcup \lbrace W^{u}_{\delta ,(\omega ,y)}\cap \lbrace z:|z-x_{\ast }|=2rr_2\rbrace : y\in A_{\ast } \ \text{such that} \ (\omega ,y)\in \Gamma _{\ell _0} \rbrace ,\) 
and let
\(\partial \mathcal {S}^r:=\bigcup \lbrace \lbrace \omega \rbrace \times \partial \mathcal {S}^{r}_{\omega }: \exists y\in A_{\ast } \ \text{with} \ (\omega ,y)\in \Gamma _{\ell _0}\rbrace .\) 
For fixed \((\omega ,x)\in \widetilde{\mathcal {S}}\cap \Gamma \) .
Since \(\lbrace \omega \rbrace \times \partial (\eta (\omega ,x)_{\omega })\subset \cup _{n\ge 0} \Phi ^{n} (\partial (\mathcal {S}^r))\) , we have that for \(0<\varepsilon <{\delta }^{\prime }_{1}\)  (\({\delta }^{\prime }_{1}\)  be as in Theorem REF ), if \(f^{-n}_{\omega }W^u_{\varepsilon }(\omega ,x)\cap (\partial (\mathcal {S}^r))_{\theta ^{-n}\omega }=\varnothing \)  then \(W^u_{\varepsilon }(\omega ,x)\subset \eta (\omega ,x)_{\omega }\) .
To guarantee that, we needed that for every \(y\in W^u_{\varepsilon }(\omega ,x)\)  and every \(n\in \mathbb {N}\) ,
\(|f^{-n}_{\omega }y-f^{-n}_{\omega }x|<|(2rr_2-|f^{-n}_{\omega }x-x_{\ast }|)|.\) 
For every \(y\in W^u_{\varepsilon }(\omega ,x)\) , by Theorem REF  (iii) we have
\(3|f^{-n}_{\omega }y-f^{-n}_{\omega }x|\le |f^{-n}_{\omega }y-f^{-n}_{\omega }x|^{\prime }_{\Phi ^{-n}(\omega ,x)}\le |x-y|^{\prime }_{(\omega ,x)}e^{\frac{-n\lambda }{2}}.\) 
Therefore, we need to take \(\varepsilon (\omega ,x)\)  small enough such that
\(\varepsilon <\min _{n\in \mathbb {N}}\lbrace {\delta }^{\prime }_1, \frac{1}{3}\ell (\omega ,x)e^{\frac{n\lambda }{2}} |(2rr_2-|f^{-n}_{\omega }x-x_{\ast }|)|\rbrace .\) 
Take \(r\in (1/2,1)\) , then the following subset
\(\lbrace r:\sum _{n=0}^{\infty }\mu (\lbrace (\omega ,x): |(|x-x_{\ast }|-2rr_2)|<e^{\frac{-n\lambda }{2}}\rbrace )<\infty \rbrace \) 
have full Lebesgue measure on the interval \((1/2,1)\)  by standard estimations (see [7]}, [4]} for details). Since there are most countable choice of \(r\)  such that \(\mu (\partial \mathcal {S}^r)>0\) , one can choose \(r\in (1/2,1)\)  such that \(r\)  belongs to the set (REF ), and \(\mu (\partial \mathcal {S}^r)=0\) .
Moreover, by \(\mu \)  is \(\Phi \)  invariant, we also have
\(\sum _{n=0}^{\infty }\mu (\lbrace (\omega ,x): |(|f^{-n}_{\omega }x-x_{\ast }|-2rr_2)|<e^{\frac{-n\lambda }{2}}\rbrace )<\infty .\) 
By a Borel-Cantelli type argument, for \(\mu \) -almost every \((\omega ,x)\)  we have
\( |(|f^{-n}_{\omega }x-x_{\ast }|-2rr_2)|\ge e^{\frac{-n\lambda }{2}}\) 
except for a finite number of integers \(n\in \mathbb {N}\) .
So, one can choose \(\varepsilon (\omega ,x)>0\)  such that \(W^u_{\varepsilon }(\omega ,x)\subset \eta (\omega ,x)_{\omega }\) .
Item (a) be proved.
It follows from Lemma REF  the sequence of functions
\((\omega ,x) \mapsto \nu _{(\omega ,x)}(E_{\omega }\cap (\vee _{i=0}^{n-1}\xi ^{i}(\omega ,x))_{\omega }) \quad \text{for} \ n\in \mathbb {N}\) 
are measurable on \(\cup ^{n-1}_{i=0 }\Phi ^{i}(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast }))\) , and decreasing for \(n\) . Therefore, we have
\((\omega ,x) \mapsto \nu _{(\omega ,x)}(E_{\omega }\cap \eta (\omega ,x)_{\omega })=\lim _{n\rightarrow \infty }\nu _{(\omega ,x)}(E_{\omega }\cap (\vee _{i=0}^{n}\eta ^{i}(\omega ,x))_{\omega })\) 
is measurable on a full \(\mu \) -measure set \(\cup ^{\infty }_{n=0}\Phi ^{i}(\Gamma _{\ell _0}\cap (\Omega \times A_{\ast }))\) . Item (c) be proved.
Item (d) follows from (REF ) and Lemma REF  directly.

SRB measures implies entropy formula
Since Ruelle's inequality [9]} (see also [10]} for the random case on Banach spaces) tells us that
\(h({\mu }^{\prime }|\mathbb {P})\le \int \sum _{i} m_i\lambda ^{+}_{i}d{\mu }^{\prime }\) 
for any \(\Phi \) -invariant probability measure \({\mu }^{\prime }\)  projection to \(\mathbb {P}\) .
Then, it suffices to show that if \(\mu \)  is a random SRB measure the reverse inequality holds.
We assume that \(\mu \)  is ergodic here.
Let \(\eta \)  be constructed in proposition REF .
Recall that \(\eta _0:=\lbrace \lbrace \omega \rbrace \times \mathcal {B}\rbrace _{\omega \in \Omega }\) .
Since \(\eta \)  is \(\Phi \) -decreasing and \(\eta _0\le \eta \) , we have
\(h(\mu |\mathbb {P},\eta )&=\lim _{n\rightarrow \infty } \frac{1}{n}H_{\mu }(\vee _{i=0}^{n}\Phi ^{-1}\eta | \eta _0)\\&=\frac{1}{n}H_{\mu }(\Phi ^{-n}\eta |\eta )=H_{\mu }(\Phi ^{-1}\eta |\eta ).\) 

It suffices to show that
\(H_{\mu }(\Phi ^{-1}\eta |\eta )=\int \log J^{u}(\omega ,x)d\mu (\omega ,x)=\sum m_i\lambda _{i}^{+},\) 
where \(J^{u}(\omega ,x)=\det (Df_{\omega }(x)|E^{u}(\omega ,x))\) .
Note that \(\log J^{u}\in L^1(\mu )\) , the second equality follows immediately from the Birkhoff ergodic theorem and Proposition REF .
By Proposition REF  (c), we let \(\nu \)  be the \(\sigma \) -finite measure defined by
\(\nu (K)=\int v_{(\omega ,x)}((\eta (\omega ,x) \cap K)_{\omega }) d\mu (\omega ,x).\) 
By assumption, we have \(\mu \ll \nu \) . Let \(\rho =\frac{d\mu }{d\nu }\) , then for \(\mu \) -almost every \((\omega ,x)\)  one has
\(\rho (\omega ,y)=\frac{d\mu _{\eta (\omega ,x)}}{d\nu _{(\omega ,x)}}(y) \quad \text{for} \ \nu _{(\omega ,x)}\text{-almost every}\ y\in \eta (\omega ,x)_{\omega }.\) 
Recall that the information function \(I(\Phi ^{-1}\eta |\eta )\)  is
\(I(\Phi ^{-1}\eta |\eta )(\omega ,x)=-\log \mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega }).\) 
Let \(L(\omega ,x)=\rho (\omega ,x)(J^{u}(\omega ,x))^{-1} (\rho (\Phi (\omega ,x)))^{-1}\) . For every measurable set \(E\subset \mathcal {B}\) .
On the one hand, since \(\eta \)  is \(\Phi \) -decreasing, one has
\(\mu _{(\Phi ^{-1}\eta )(\omega ,x)}(E)&=\dfrac{\mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega }\cap E) }{\mu _{\eta (\omega ,x)}((\Phi ^{-1}\eta )(\omega ,x)_{\omega })}\\&=e^{I(\Phi ^{-1}\eta |\eta )(\omega ,x)}\int _{((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E} \rho (\omega ,y) d\nu _{(\omega ,x)}(y).\) 

On the other hand, by \(\mu \)  is \(\Phi \) -invariant, we have for \(\mu \) -almost every \((\omega ,x)\)
\(\mu _{(\Phi ^{-1}\eta )(\omega ,x)}(E)&=\mu _{\eta (\Phi (\omega ,x))}(f_{\omega }E)\\&=\int _{f_{\omega }(((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E)} \rho (\theta (\omega ),z) d\nu _{\Phi (\omega ,x)}(z)\\&=\int _{f_{\omega }(((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E)} \rho (\theta (\omega ),z) J^{u}(\omega ,f^{-1}_{\theta \omega }(z)) d((f_{\omega })_{\ast }\nu _{(\omega ,x)})(z)\\&=\int _{((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\cap E} \rho (\Phi (\omega ,y)) J^{u}(\omega ,y) d\nu _{(\omega ,x)}(y),\) 

where we use the change of variables formula for induced measures on finite-dimensional manifolds in the third equality
(recall \(f^{-1}_{\theta \omega }(y)\)  is the unique point such that \(f_{\omega }(f^{-1}_{\theta \omega }(y))=y\) ).
Consequently, by \(E\)  is arbitrary and \( I(\Phi ^{-1}\eta |\eta )(\omega ,y)= I(\Phi ^{-1}\eta |\eta )(\omega ,x)\)  for any
\(y\in ((\Phi ^{-1}\eta )(\omega ,x))_{\omega }\) , we get that for \(\mu \) -almost every \((\omega ,x)\)
\(I(\Phi ^{-1}\eta |\eta )(\omega ,y)=-\log L(\omega ,y) \quad \text{for} \ \nu _{(\omega ,x)}\text{-almost every}\ y\in ((\Phi ^{-1}\eta )(\omega ,x))_{\omega }.\) 
Therefore, for \(\mu \) -almost every \((\omega ,x)\)  one has
\(I(\Phi ^{-1}\eta |\eta )(\omega ,x)=\log J^{u}(\omega ,x)+\log \dfrac{\rho (\Phi (\omega ,x))}{\rho (\omega ,x)}.\) 
Since \(\log J^{u}\in L^1(\mu )\)  and \(I(\Phi ^{-1}\eta |\eta )(\omega ,x)\ge 0\) , one has \(\log ^{-} \frac{\rho \circ \Phi }{\rho }\in L^1(\mu )\) . Therefore, by Lemma REF  one has \(\int \log \frac{\rho \circ \Phi }{\rho }d\mu =0\) .
So, we have
\(H_{\mu }(\Phi ^{-1}\eta |\eta )=\int \log J^{u}(\omega ,x)d\mu (\omega ,x).\) 
This completes the proof.

Entropy formula implies SRB measures
We assume that \((\Phi ,\mu )\)  satisfies (H1)-(H5) and \(\mu \)  is ergodic. Let the partition \(\eta \)  be constructed in Proposition REF .
We first prove the following:
Lemma 5.2 
Let \((\Phi ,\mu )\)  with a positive Lyapunov exponent for \(\mu \) -almost everywhere, and let \(\eta \)  be as in Proposition REF . Assume
\(h(\mu |\mathbb {P},\eta )=\int \log J^{u} d\mu ,\) 
then we have \(\mu \)  is a Random SRB measure.

Let \(\eta ,\xi , \mathcal {S}\)  and \(\widetilde{\mathcal {S}}\)  be as in Proposition REF .
For \((\omega ,x)\in \mathcal {S}\) . Recall the Lipschitz-continuous function \(z \mapsto \Delta ^{\omega }(x,z)\)  on \(\xi (\omega ,x)_{\omega }\)  which defined by
\(z\mapsto \Delta ^{\omega }(x,z):= \prod _{n=1}^{\infty } \dfrac{J^{u}(\Phi ^{-n}(\omega ,x))}{J^{u}(\Phi ^{-n}(\omega ,z))}.\) 
And the function \(q(\omega ,z)\)  on \(\mathcal {S}\)  defined by
\(q(\omega ,z):=\dfrac{\Delta ^{\omega }(x,z)}{\int _{\xi _{\omega }(x)} \Delta ^{\omega }(x,y) d\nu _{(\omega ,x)}(y)}, \quad z\in \xi (\omega ,x)_{\omega }.\) 
Since \(\eta \)  is subordinate to \(W^{u}(\omega ,x)\) ,
for every \((\omega ,x)\)  there exists \(n\ge 0\)  such that \(f^{-n}_{\theta ^{-n}\omega }(\eta (\Phi ^{-n}(\omega ,x))_{\theta ^{-n}\omega })\)  contained in a leaf of \(\mathcal {S}_{\omega }\) .
By assumption (H1) and Theorem REF , \(f^{n}_{\theta ^{-n}\omega }\)  is a \(C^2\)  embedding on each unstable leaf.
Then, we have that for \(\mu \) -almost every \((\omega ,x)\) , the map \(z\mapsto \Delta ^{\omega }(x,z)\)  is Lipschitz-continuous and bounded on \(\eta (\omega ,x)_{\omega }\) .
By Proposition REF  (c), for \(\mu \) -almost every \((\omega ,x)\)  we have \(\nu _{(\omega ,x)}(\eta (\omega ,x)_{\omega })>0\) .
Therefore, the following function
\(p(\omega ,z):=\dfrac{\Delta ^{\omega }(x,z)}{\int _{\eta (\omega ,x)_{\omega }} \Delta ^{\omega }(x,y) d\nu _{(\omega ,x)}(y)}, \ z\in \eta (\omega ,x)_{\omega },\) 
is well defined for \(\mu \) -almost every \((\omega ,x)\) .
Suppose we have that \(\mu _{\eta (\omega ,x)}\ll \nu _{(\omega ,x)}\)  for \(\mu \) -almost \((\omega ,x)\) .
Then there exist some function \(\rho \)  such that \(d\mu _{\eta (\omega ,x)}=\rho \nu _{(\omega ,x)}\)  for \(\nu _{(\omega ,x)}\) -almost every
\(y\in \eta (\omega ,x)_{\omega }\) .
By the argument in the proof of Theorem A, the function \(\rho \)  must satisfy \(\int _{\eta (\omega ,x)_{\omega }}\rho d\nu _{(\omega ,x)}=1\)  and \(\rho (\Phi (\omega ,y))J^{u}(\omega ,y)\rho (\omega ,y)^{-1}\)  must be constant on \((\Phi ^{-1}\eta )(\omega ,x)_{\omega }\) .
So, one can guess that \(\rho =p\) .
The function \(p\)  is measurable by Lemma REF  and Lemma REF  with standard arguments.
So, we can define a probability measure \(\upsilon \)  on \(\widetilde{\mathcal {S}}\)  by
\(\upsilon (K)=\int (\int _{(K\cap \eta (\omega ,x))_{\omega }} p(\omega ,y) dv_{(\omega ,x)}(y))d\mu (\omega ,x) \) 
for each \(K\)  measurable.
By construction, the measure \(\upsilon \)  and \(\mu \)  project to the same measure on the quotient space \(\widetilde{\mathcal {S}}/\eta \) , and the conditional measure of \(\upsilon \)  on each element of \(\eta \) , which we denote by \(\upsilon _{\eta (\omega ,x)}\) , is absolutely continuous with respect to \(\nu _{(\omega ,x)}\)  with the densities function \(p\) .
It suffices to prove that \(\upsilon =\mu \) .
For each \(n\in \mathbb {N}\) , by the construction we have
\(\upsilon _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })=\dfrac{\int _{(\Phi ^{-n}\eta )(\omega ,x)_{\omega }} \Delta ^{\omega }(x,y)d\nu _{(\omega ,x)}(y)}{\int _{\eta (\omega ,x)_{\omega }} \Delta ^{\omega }(x,y)d\nu _{(\omega ,x)}(y)}.\) 
By the change of variables formula (recall \(f^{-n}_{\theta ^n\omega }(y)\)  is the unique point such that
\(f^n_{\omega }(f^{-n}_{\theta ^n\omega }(y))=y\) ),
\(&\int _{(\Phi ^{-n}\eta )(\omega ,x)_{\omega }} \Delta ^{\omega }(x,z)d\nu _{(\omega ,x)}(z)\\=&\int _{f^{-n}_{\theta ^n\omega }(\eta (\Phi ^{n}(\omega ,x))_{\theta ^{n}\omega })} \Delta ^{\omega }(x,z)d\nu _{(\omega ,x)}(z)\\=&\int _{\eta (\Phi ^{n}(\omega ,x))_{\theta ^{n}\omega }} \Delta ^{\omega }(x,f^{-n}_{\theta ^n\omega }(y))d((f^n_{\omega })_{\ast }\nu _{(\omega ,x)})(y)\\=&\int _{\eta (\Phi ^{n}(\omega ,x))_{\theta ^{n}\omega }} \dfrac{\Delta ^{\omega }(x,f^{-n}_{\theta ^n\omega }(y))}{\det (Df^n_{\omega }(f^{-n}_{\theta ^n\omega }(y))|E^{u}(\omega ,f^{-n}_{\theta ^n\omega }(y)))}d\nu _{\Phi ^{n}(\omega ,x)}(y)\\=&\dfrac{1}{\det (Df^n_{\omega }(x)|E^{u}(\omega ,x))} \int _{\eta (\Phi ^{n}(\omega ,x))_{\theta ^{n}\omega }} \Delta ^{\theta ^n\omega }(f^n_\omega x,y)d\nu _{\Phi ^{n}(\omega ,x)}(y).\) 

Let
\(L(\omega ,x):=\int _{\eta (\omega ,x)_{\omega }} \Delta ^{\omega }(x,y)d\nu _{(\omega ,x)}(y)\) , then we have
\(\upsilon _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })=\dfrac{L(\Phi ^n(\omega ,x))}{L(\omega ,x)}\cdot \dfrac{1}{\det (Df^n_{\omega }(x)|E^{u}(\omega ,x))}.\) 
Using the same argument in the proof of Theorem A, for each \(n>0\)  we have
\(\int -\log \upsilon _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })d\mu (\omega ,x)=\int \log \det (Df^n_{\omega }(\omega ,x)|E^{u}(\omega ,x))d\mu (\omega ,x).\) 
Note that the main assumption of the lemma tells us
\(\begin{aligned}\int -\log \mu _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })&d\mu (\omega ,x)=H_{\mu }(\Phi ^{-n}\eta |\eta )\\=&\int \log \det (Df^n_{\omega }(\omega ,x)|E^{u}(\omega ,x))d\mu (\omega ,x).\end{aligned}\) 
Consider the \(\widetilde{\mathcal {S}}/(\Phi ^{-n}\eta )\) -measurable function
\(\psi _n(\omega ,x)=\dfrac{\upsilon _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })}{\mu _{\eta (\omega ,x)} ((\Phi ^{-n}\eta )(\omega ,x)_{\omega })},\) 
which is defined for \(\mu \) -almost everywhere.
Together with (REF ) and (REF ) we have \(\int \log \psi _n d\mu =0\) .
Let \(\mathbb {B}_n\)  be the sub-\(\sigma \) -algebra of measurable subsets that are unions of elements of \(\Phi ^{-n}\eta \) , and let \(\mu _n\)  and \(\upsilon _n\)  be the restriction of \(\mu \)  and \(\upsilon \)  on \(\mathbb {B}_n\)  respectively.
Decompose \(\upsilon _n=\upsilon _{n}^{\mu }+\upsilon ^{\perp }_n\)  where \(\upsilon _{n}^{\mu }\ll \mu _n\)  and \( \upsilon ^{\perp }_n \)  is mutually singular with \(\mu _n\)  (\(\upsilon ^{\perp }_n\)  can be strictly positive), then \(\psi _n=\frac{d\upsilon _{n}^{\mu }}{d\mu _{n}}\) .
Thus, we have \(\int \psi _n d\mu \le 1\) .
By Jensen's inequality we have
\(0=\int \log \psi _n d\mu \le \log \int \psi _n d\mu \le 0\) 
and the equality holds if and only if \(\psi _n\)  is constant for \(\mu \) -almost everywhere.
So, \(\psi _n\equiv 1\)  for \(\mu \) -almost every \((\omega ,x)\) .
This implies \(\mu \)  and \(\upsilon \)  coincide on \(\mathbb {B}_n\) . Since \(\Phi ^{-n}\eta \)  partition into points modulo \(\mu \) -measure 0, one has \( \mathbb {B}_n\nearrow \mathcal {F}\otimes \mathcal {B}\) , and so \(\mu =\upsilon \) . This completes the proof.
By Lemma REF , it suffices to show that
\(h(\mu |\mathbb {P},\eta )=h(\mu |\mathbb {P}).\) 
We will use the following lemma which needed that \(A\)  is visible:
Lemma 5.3 ([6]} Proposition 4.2) 
Let \((\Phi ,\mu )\)  be a random dynamical system where \(\mu \)  is \(\Phi \) -invariant and supported on a random compact set \(A\)  with \(\Phi (A)=A\) . For any function \(\tau :A\rightarrow (0,1)\)  with \(\log \tau \in L^1(\mu )\) .
If \(A\)  is visible, then there exists a measurable partition \(\mathcal {P}\)  of \(A\)  such that
\(h(\mu |\mathbb {P},\mathcal {P})<\infty \)  and \(\mathcal {P}(\omega ,x)_{\omega }\subset B(x,\tau (\omega ,x))\)  for \(\mu \) -almost every \((\omega ,x)\) .

Let \(\tau :\Gamma _{\ell _0}\rightarrow \mathbb {N}\)  be the first return time to \(\Gamma _{\ell _0}\) , extend \(\tau \)  to all of \(A\)  by setting \(\tau |_{A \setminus \Gamma _{\ell _0}}=0\) .
Let \(\rho _{\delta }(\omega ,x):=\delta l^{-2}_0e^{2\tau (\omega ,x)\varepsilon _0}\) .
Since \(\tau \in L^1(\mu )\)  we have \(\log \rho _{\delta }\in L^1(\mu )\) .
Using Lemma REF , there exists a partition \(\mathcal {P}\)  of \(A\)  with finite conditional entropy and \(\mathcal {P}(\omega ,x)_{\omega }\subset B(x,\rho _{\delta }(\omega ,x))\) .
Let \(\mathcal {P}^{+}:=\vee _{n\ge 0}\Phi ^{n}\mathcal {P}\) .
It is straightforward to check that for \(\mu \) -almost every \((\omega ,x)\) , for each \(y\in \mathcal {P}^{+}(\omega ,x)_{\omega }\)  one has
\(|f_{\omega }^{-n}y-f_{\omega }^{-n}x|^{\prime }_{\Phi ^{-n}(\omega ,x)}\le \delta \ell (\Phi ^{-n}(\omega ,x))^{-1}.\) 
Then, by Lemma REF  one has
\(\mathcal {P}^{+}(\omega ,x)_{\omega }\subset W^{u}_{\delta }(\omega ,x)\)  (note that we assume \(E^{c}=\lbrace 0\rbrace \) ).
For each \(r>0\) . Choose a countable partition \(\mathcal {P}_0\)  of \(A\)  with finite conditional entropy such that
\(h(\mu |\mathbb {P})<h(\mu |\mathbb {P},\mathcal {P}_0)-r\) .
Let \(\mathcal {Q}:= \mathcal {P}\vee \lbrace \widetilde{\mathcal {S}}\cap A,A-\widetilde{\mathcal {S}}\rbrace \vee \mathcal {P}_0\) . Then, we also have \(h(\mu |\mathbb {P})<h(\mu |\mathbb {P},\mathcal {Q})-r\) . Let \( \mathcal {Q}^{+}=\vee _{n\ge 0}\Phi ^{n}\mathcal {Q}\) .
Use the fact that \(\mathcal {P}^{+}(\omega ,x)_{\omega }\subset W^{u}_{\delta }(\omega ,x)\)  and the construction of \(\eta \)  we can check that \(\eta (\omega ,x)_{\omega }\supset \mathcal {Q}^{+}(\omega ,x)_{\omega }\)  for \(\mu \) -almost every \((\omega ,x)\) .
This implies that \(\eta \le \mathcal {Q}^{+}\vee \eta _0\) .
Then, we have
\(h(\mu |\mathbb {P},\eta )&=\frac{1}{n} H_{\mu }(\Phi ^{-n}\eta |\eta )\\&\ge \frac{1}{n} H_{\mu }(\vee _{j=0}^{n} \Phi ^{-j}\mathcal {Q}|\eta )-\frac{1}{n} H_{\mu }(\vee _{j=0}^{n} \Phi ^{-j}\mathcal {Q}|\Phi ^{-n}\eta )\\&\ge \frac{1}{n} H_{\mu }(\vee _{j=0}^{n} \Phi ^{-j}\mathcal {Q}|\mathcal {Q}^{+}\vee \eta _0)-\frac{1}{n} H_{\mu }(\vee _{j=0}^{n} \Phi ^{-j}\mathcal {Q}|\Phi ^{-n}\eta )\\&\ge h(\mu |\mathbb {P},\mathcal {Q})-\frac{1}{n} H_{\mu }(\vee _{j=0}^{n} \Phi ^{-j}\mathcal {Q}|\Phi ^{-n}\eta ),\) 

the second term in above can be shown to be \(<r\)  for large \(n\)  since \(\vee _{n\ge 0}\Phi ^{-n} \eta \)  partition into points \(\mu \) -modulo 0. Therefore, we have that
\(h(\mu |\mathbb {P},\eta )\ge h(\mu |\mathbb {P},\mathcal {Q})-r>h(\mu |\mathbb {P})-2r.\) 
By arbitrarily of \(r\) ,
we have \(h(\mu |\mathbb {P},\eta )=h(\mu |\mathbb {P})\) .
This completes the proof of Theorem B in the ergodic case.

The non-ergodic case
The case when \(\mu \)  is not ergodic can be proved in the same way with the following modifications.
The non-ergodic case of the Multiplicative Ergodic Theorem is as follows: fixed a measurable function \( \lambda _{\alpha }\)  with \(0>\lambda _{\alpha }>l_{\alpha }\) .
Then there is a measurable function \(r:\Gamma \rightarrow \mathbb {N}\)  such that for every \((\omega ,x)\in \Gamma \) , there are \(r(\omega ,x)\)  Lyapunov exponents
\(\lambda _1(\omega ,x)\ge \cdots \lambda _{r(\omega ,x)}(\omega ,x)(> \lambda _{\alpha }(\omega ,x))\) 
and a unique splitting
\(\mathcal {B}=E_1(\omega ,x)\oplus \cdots E_{r(\omega ,x)}(\omega ,x) \oplus F(\omega ,x)\) 
with respect to the properties (a)-(d) in Theorem REF  hold.
Moreover, \(\lambda _{i}(\omega ,x)\)  and \(m_i(\omega ,x):=\dim E_{i}(\omega ,x)\)  are measurable functions on \(\lbrace (\omega ,x):r(\omega ,x)\ge i\rbrace .\) 
The limit in Proposition REF  also exists for \(\mu \) -almost everywhere, but the equality is
\(\int \lim _{n\rightarrow \infty } \frac{1}{n}\log \det (Df^{n}_{\omega }(x)|\bigoplus _{j=1}^{k}E_{i_j}(\omega ,x)) d\mu (\omega ,x)\\=\int m_i(\omega ,x)\lambda _{i_j}(\omega ,x)d\mu (\omega ,x).\) 
The unstable, center, and stable subspace in Section REF  are also defined, and let \(\lambda ^{\pm }(\omega ,x)\)  be as before, which are measurable functions with positive (negative) values but not be bounded away from 0.
For \(m,n\in \mathbb {N}\)  and \(p,q\in \mathbb {Z}^{+}\) , let
\(\Gamma (m,n,p,q):=\lbrace (\omega ,x):&\dim E^{u}(\omega ,x)=m, \dim E^{c}(\omega ,x)=n; \\&\lambda ^{+}(\omega ,x)\ge \frac{1}{p}, \lambda ^{-}(\omega ,x)\le -\frac{1}{q}\rbrace .\) 

Then, each \(\Gamma (m,n,p,q)\)  is either empty, or it is \(\Phi \) -invariant, and
\(\Gamma =\bigcup _{m,n,p,q}\Gamma (m,n,p,q).\) 
Therefore, the results in Section  and Section  are also holding on \(\Gamma (m,n,p,q)\) .
Note that \(\Gamma (m,n,p,q)\)  is not disjoint, we let
\(\bar{\Gamma }(m,n,p,q)=\Gamma (m,n,p,q)\setminus \Gamma (m,n,p-1,q-1).\) 
Then, \(\bar{\Gamma }\)  is disjoint, and \(\Gamma \)  is a countable union of \(\bar{\Gamma }(m,n,p,q)\) .
We can see that \(\bar{\Gamma }\)  is an alternative to ergodic decomposition [1]}.
In non-ergodic cases, it may happen that the set \(\widetilde{\mathcal {S}}\)  which is defined in Section  is not a full \(\mu \) -measure set. However, it is no problem that since there is at most a countable union of disjoint sets of the form \(\widetilde{\mathcal {S}}\)  will cover a full \(\mu \) -measure set.
